Download probabilistic methods for location estimation in wireless

PROBABILISTIC METHODS FOR
LOCATION ESTIMATION IN
WIRELESS NETWORKS
Petri Kontkanen, Petri Myllymäki, Teemu Roos,
Henry Tirri, Kimmo Valtonen and Hannes Wettig
June 7, 2004
HIIT
TECHNICAL
REPORT
2004–3
PROBABILISTIC METHODS FOR LOCATION ESTIMATION
IN WIRELESS NETWORKS
Petri Kontkanen, Petri Myllymäki, Teemu Roos, Henry Tirri, Kimmo Valtonen
and Hannes Wettig
Helsinki Institute for Information Technology HIIT
Tammasaarenkatu 3, Helsinki, Finland
PO BOX 9800
FI-02015 TKK, Finland
http://www.hiit.fi
HIIT Technical Reports 2004–3
ISSN 1458-9478
URL: http://cosco.hiit.fi/Articles/hiit-2004-3.pdf
c 2004 held by the authors
Copyright °
NB. The HIIT Technical Reports series is intended for rapid dissemination of results
produced by the HIIT researchers. Therefore, some of the results may also be later
published as scientific articles elsewhere.
In: EMERGING LOCATION AWARE BROADBAND WIRELESS ADHOC
NETWORKS, edited by Rajamani Ganesh, Sastri Kota, Kaveh Pahlavan and
Ramón Agustí. Kluwer Academic Publishers, 2004.
Chapter 11
PROBABILISTIC METHODS FOR LOCATION
ESTIMATION IN WIRELESS NETWORKS
Petri Kontkanen, Petri Myllymäki, Teemu Roos, Henry Tirri, Kimmo
Valtonen and Hannes Wettig
Complex Systems Computation Group, Helsinki Institute for Information Technology,
University of Helsinki & Helsinki University of Technology, P.O.Box 9800, 02015 HUT,
Finland
Abstract:
Probabilistic modeling techniques offer a unifying theoretical framework for
solving the problems encountered when developing location-aware and
location-sensitive applications in wireless radio networks. In this paper we
demonstrate the usefulness of the probabilistic modelling framework in
solving not only the actual location estimation (positioning) problem, but also
many related problems involving pragmatically important issues like
calibration, active learning, error estimation and tracking with history. Some
interesting links between positioning research done in the area of robotics and
in the area of wireless radio networks are also discussed.
Key words:
Location estimation, probabilistic modeling, wireless networks
1.
INTRODUCTION
The location of a mobile terminal can be estimated using radio signals
transmitted or received by the terminal. The problem is called with various
names such as location estimation, geolocation, location identification,
location determination, localization, and positioning. The traditional,
geometric approach to location estimation is based on angle and distance
estimates from which a location estimate is deduced using standard
geometry. Instead of the geometric approach, we consider the probabilistic
approach which is based on probabilistic models that describe the
dependency of observed signal properties on the location of the terminal, and
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the motion of the terminal. The models are used to estimate the terminal's
location when signal measurements are available.
The feasibility of the probabilistic approach in the context of wireless
networks has already been demonstrated to some extent in a number of
recent papers (Castro et al., 2001; Ladd et al., 2002, Roos et al., 2002a, Roos
et al., 2002b; Schwaighofer et al., 2003; Youssef et al., 2003). Probabilistic
methods have also been extensively used in robotics where they provide a
natural way to handle uncertainty and errors in sensor data (Smith et al.,
1990; Burgard et al., 1996; Thrun, 2000). In a recent survey (Thrun, 2003),
Sebastian Thrun summarizes the central role of probabilistic methods in
robotic mapping as follows:
‘‘Virtually all state-of-the-art algorithms for robotic mapping in the
literature have one common feature: They are probabilistic. […] The
reason for the popularity of probabilistic techniques stems from the fact
that robot mapping is characterized by uncertainty and sensor noise. ’’
Many of the probabilistic methods developed in the robotics community, in
particular those related to mapping, location estimation and tracking, are also
applicable in the context of wireless networks. In the following we discuss
selected topics in probabilistic location estimation, many of which are wellknown in probabilistic modeling, but have received relatively little attention
in the domain of wireless networks.
We focus primarily on wireless local area networks, WLANs, but most of
the ideas and concepts are applicable to many other wireless networks as
well, including those based on GSM/GPRS, CDMA or UMTS standards.
The rest of te paper is organized as follows: In Section 2 we discuss
calibration, the process of obtaining a model of the signal properties at
various locations. The actual location estimation and tracking phase
following calibration is considered in Sections 3 and 6. Issues related to the
optimal choice of calibration measurements are discussed in Section 4. In
many cases, it is useful to complement a location estimate with information
on its accuracy; in Section 5 we describe methods for error estimation and
visualization. Conclusions are summarized in Section 7.
2.
CALIBRATION
In order to obtain a positioning model, we need to estimate the
distribution of the signal properties, e.g., signal strength, as measured by the
device to be localized for the various locations in consideration. This has
traditionally been done using knowledge of radiowave propagation. Several
propagation prediction or cell planning tools are available for this purpose
11. Probabilistic Methods for Location Estimation in Wireless
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3
(Andersen et al., 1995; Wölfle and Landstorfer, 1999). We adopt an
empirical approach, i.e. we estimate the required distributions from
calibration data gathered at different locations in consideration.
Experimental studies suggest that propagation methods are not competitive
against empirical models in terms of positioning accuracy due to
insufficiently precise signal models (Bahl and Padmanabhan, 2000, Roos et
al., 2002b).
Consider a finite set of calibration points l, which are labelled by their x
and y coordinates (and possibly a third coordinate z or other additional
information). For each calibration point we gather calibration data, i.e., a
number of observation vectors o to estimate the distribution of signal
properties from. For a discussion on how this can and should be done for a
given set of calibration points and observations see (Castro et al., 2001; Roos
et al., 2002a; Schwaighofer et al., 2003; Youssef et al., 2003). When we then
want to position a device with current signal readings o, we calculate the
probabilities p(l | o) for each possible location l using the Bayes rule and the
distributions estimated from the calibration data as described in Section 3.
For simplicity, we assume that the set of possible locations can be
considered equal to the set of calibration points. If this is not the case, e.g.,
when a continuous location variable is used, we need to interpolate in order
to obtain a distribution of the signal properties at locations from which no
calibration data is available. Note that without interpolation we can model
only a finite—and for practical reasons preferably not too large—set of
possible locations. We can then determine for example in which room a
device is located (with certain probability), but have no probabilities
associated to locations inbetween the calibration points.
But how should we choose the set of possible locations and how should
we collect calibration data? A simple solution is to use a probability grid
(Burgard, 1996), dividing the positioning space into cells of some size, e.g.,
1m × 1m. In order to obtain a distribution of the signal properties at each
grid point without interpolation, one then needs to collect a sufficient
number of training vectors at each grid point. However, it may be
impractical to remain at each grid point for the time it takes to gather enough
data—and favorably move around in order to capture variance due to
orientation and within the area of the cell—before moving on to the next
cell.
A more convenient way is to gather data vectors continuously just
walking (or driving) around. We only need to record the time label of each
observation and the time labels and coordinates of those locations at which
the calibrator changes direction and/or speed. This way we quickly obtain a
large number of observations equipped with their exact location. However,
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Chapter 11
we (usually) get only one observation per location, which does not suffice to
reliably estimate the distribution of signal properties. Furthermore it is
computationally problematic to deal with such a large number of possible
locations in a model; note that when a device supplies us with an observation
vector every 500ms, a calibration round of an hour already yields up to 7200
locations.
A natural way of dealing with this situation is to group the locations into
clusters (Youssef, 2003). Each cluster should consist of a sufficient number
of vectors to supply a good estimate of the signal properties in its area, and
as its location we may take—for example—the center of gravity of its
measurements' positions. An interesting and theoretically appealing way to
produce such clustering is given by the principle of Minimum Description
Length (MDL) in its most recent form, the Normalized Maximum Likelihood
(NML) (Rissanen, 1996), for details see (Kontkanen et al., 2004). Figure 1
shows such clustering of a calibration tour. Note, that there is no need to
decide on the number of clusters in advance, the algorithm will choose as
many as can reliably be distinguished from the data collected.
Figure 11-1. NML clustering of signal data collected continuously along a calibration tour.
Each circle is represents one vector of measurements gathered at its position, the different
clusters are colour-coded.
3.
LOCATION ESTIMATION
After the calibration phase we have, for any given location l, a
probability distribution p(o | l) that assigns a probability (density) for each
measured signal vector o. By application of the Bayes rule, we can then
11. Probabilistic Methods for Location Estimation in Wireless
Networks
5
obtain the so called posterior distribution of the location (Roos et al.,
2002b):
p(l | o) = p(o | l) p(l) / p(o) = p(o | l) p(l) / (
Σl' ∈ L p(o | l') p(l') ),
where p(l) is the prior probability of being at location l before knowing the
value of the observation variable, and the summation goes over the set of
possible location values, denoted by L. If the location variable is continuous,
the sum is replaced by the corresponding integral.
The prior distribution p(l) gives a principled way to incorporate
background information such as personal user profiles and to implement
tracking as described in Section 6. In case neither user profiles nor a history
of measured signal properties allowing tracking are available, one can
simply use a uniform prior which introduces no bias towards any particular
location. As the denominator p(o) does not depend on the location variable l,
it can be treated as a normalizing constant whenever only relative
probabilities or probability ratios are required.
The posterior distribution p(l | o) can be used to choose an optimal
estimator of the location based on whatever loss function is considered to
express the desired behavior. For instance, the squared error penalizes large
errors more than small ones, which is often useful. If the squared error is
used, the estimator minimizing the expected loss is the expected value of the
location variable:
E[l|o]=
Σl ∈ L l p(l | o),
assuming that the expectation of the location variable is well defined, i.e.,
the location variable is numerical. Location estimates, such as the
expectation, are much more useful if they are complemented with some
indication about their precision. We discuss error estimates in Section 5
below.
The presented probabilistic approach can be contrasted with the more
traditional, geometric approach to location estimation used in methods such
as angle-of-arrival (AOA), time-of-arrival (TOA), and time-difference-ofarrival (TDOA). In the geometric approach the signal measurements are
transformed into angle and distance estimates from which a location estimate
is deduced using standard geometry. One of the drawbacks of the geometric
approach is that there is no principled way to deal with the incompatibility of
the angle and distance estimates caused by measurement errors and noise.
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On the other hand, the geometric approach is usually computationally very
efficient.
4.
ACTIVE LEARNING
In Section 2 we only considered the problem of obtaining a model of the
signal properties given training data collected from known locations. The
resulting model is strongly dependent on where and how much training data
is collected. Obviously, the training data should not leave large areas
uncovered or otherwise there would be no way to reliably infer the signal
properties in such areas. Also, for various reasons, for some areas the signal
model is required to be more accurate than in general, in order to achieve
accurate location estimation. For instance, two distinct locations may be
roughly similar in terms of signal properties so that they can be told apart
only by a small margin. In such areas, more extensive calibration is required.
In practice, if it is possible to collect a large amount of training data, a
reasonable calibration result is obtained by collecting training data roughly
uniformly from each location. Areas where the signal properties are expected
to vary within small distances due to, for instance, large obstacles, may be
better covered with relatively higher density, whereas large open areas where
the signal is likely to be constant, can be left with less attention. In case
extensive calibration is costly or otherwise impossible, it becomes critical to
choose the calibration points as well as possible. The problem of choosing
optimal actions in order to reduce uncertainty has been studied in the
robotics literature under the name robotic exploration. In general, optimal
decision strategies are intractable and various heuristics are used (Burgard et
al., 2000; Thrun, 2003).
A practical method for locating potentially useful candidates for new
calibration points is based on the estimate of the future expected error. This
estimate is calculated by summing over all possible future observations o:
E [err | l ] =
Σo E [err | l,o] p(o),
where l is the calibration point candidate, and E[err | l,o] is the expected
error:
E [err | l,o ] =
Σ l' ∈ L p(l’ | o) d(l’,l),
for the preferred distance function d.
11. Probabilistic Methods for Location Estimation in Wireless
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7
The candidate points l can be chosen by using a tight grid. For example,
the grid spacing could be approximately one meter. One or more grid points
with a high expected error, or points surrounded by several such grid points,
are then used as new calibration points. If the dimensionality of the
observation vector o is so high that the summing over all o as above is not
feasible, the sum can be approximated by sampling. An ever simpler
approach is to use the calibration data as the set over which the sampling is
performed, in which case one only needs to sum over the calibrated
observations.
To implement the method based on equations above, one needs to
determine the probability distribution or density over the future observations.
In practice, it has to be approximated from the calibration data. One possible
approximation method is as follows. When computing E[err | l ] for some
location l, one replaces the p(o) by the probability distribution based on the
past observations made at the calibration point closest to l. The efficiency of
the method can then be further improved by approximating E[err | l,o] by
d(l*, l), where l* is the point estimate produced by the positioning system
after seeing observation o.
5.
ERROR ESTIMATION AND VISUALIZATION
In order to visualize the uncertainty associated with the location, we
assume that we have a probability distribution p, either a probability mass
function or a density, which describes the uncertainty about the actual
location. In addition to reporting a point estimate—here taken to be the
expected value—we can visualize the uncertainty related to distribution p.
This can be done, for instance, by drawing an ellipse centered at the
expected location such that the orientation and size of the ellipse describes
the uncertainty of the location estimate as well as possible.
As a first step of obtaining such an “uncertainty ellipse” one first needs to
obtain certain summary statistics from the distribution p. These statistics are,
in addition to the expectation, contained in the variance-covariance matrix.
The variance-covariance matrix describes the variance of the location in both
x and y coordinates together with the correlation of the two coordinates. The
second step is to evaluate the two eigenvectors of the variance-covariance
matrix. This is a simple exercise in linear algebra. For instance, in case the
two coordinates x and y happen to be independent in the distribution p, i.e.,
there is no correlation, the eigenvectors are parallel to the two coordinate
axes. Finally, one displays an ellipse whose axes are parallel to those given
by the two eigenvectors of the variance-covariance matrix. The lengths of
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the axes are given by the eigenvalues multiplied by a scaling constant. We
give a rule for determining the value of the scaling constant below, after we
have first discussed the interpretation of the ellipse.
One interpretation for the uncertainty ellipse is that assuming
(pretending) that the estimated density of the location is bivariate Gaussian,
the ellipse is the smallest area that contains a fixed probability mass. Given
the probability mass to be covered by the ellipse, one can obtain the
aforementioned scaling constant by taking the square root of the Chi-squared
value with two degrees of freedom. For instance, if 95 % coverage is
required, the scaling constant becomes √5.991 = 2.448. An illustration of the
error ellipse is shown in Fig. 2.
Figure 11-2. Uncertainty ellipse. Probabilities at a discrete set of locations are denoted by
circles; dark shading implies high probability. The ellipse centered at the expected location
has axes parallel to eigenvectors of the variance-covariance matrix and lengths proportional to
eigenvalues.
11. Probabilistic Methods for Location Estimation in Wireless
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Figure 11-3. Uncertainty about the estimate represented by a polar coordinate system placed
at the point estimate. Calibration points are marked by circles, colored depending on p(x,y).
The relative amount of uncertainty in each direction away from the point estimate is
visualized by the curve.
Whereas the ellipse approach shows the uncertainty about location in two
orthogonal directions with respect to the point estimate, a generalization to
an arbitrary number of directions can be obtained by mapping p(x,y) to a
polar coordinate system centered on the point estimate. In this method, the
origin is placed at the point estimate and each calibration point mapped to
the polar coordinate system a(x,y),d(x,y), where a(x,y) is the angle w.r.t. the
point estimate and d(x,y) is the distance. It is convenient to discretize both
a(x,y) and d(x,y), resulting in the case of two-dimensional space in a set of
segments that partition the space disjointly and exhaustively.
We gain a discrete two-dimensional distribution pp(a(x,y), d(x,y)) over
the location space. The curve visualizing a wanted contiguous portion of the
total mass can then be derived from pp(a(x,y), d(x,y)). Relative distances
from the origin are first determined for each sector based on expected
distances. The resulting shape describes relative probability mass in each
“direction” (sector). To represent the spread of uncertainty as well, the curve
can be scaled so that it covers a desired fraction of p p(a(x,y),d(x,y)). For a
screen shot of an implementation, see Fig. 3.
6.
TRACKING
Location estimation accuracy can be greatly improved if instead of a
single signal measurement, a series of measurements is available unless the
mobile device is moving with very high speed or the time interval between
measurements is very long. Such a series of measurements allows keeping
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track of the device's location as a function of time, also called tracking. It is
convenient to model the situation as a hidden Markov model (Rabiner, 1989)
illustrated in Fig. 4.
Figure 11-4. Hidden Markov model. State variables (white nodes) are hidden (not observed).
Observed variables are denoted by shaded nodes. Horizontal arrows correspond to transition
probabilities between successive states. Vertical arrows correspond to observation
probabilities given state.
In a hidden Markov model, the variables l1, l2, ... correspond to a
sequence of states indexed by time t. In our location estimation domain, the
state correspond to location and hence, the state sequence constitutes a
trajectory of the located device. The model also has a set of corresponding
observation variables, denoted by o1, o2, ... . Each observation variable, ot, is
assumed to be dependent only on the current location, l t. In the model in Fig.
4, the location at time t is dependent on the earlier locations only through the
previous location lt-1. Generalizations to higher order dependencies are easily
expressed in the general framework of graphical probabilistic models
(Cowell et al., 1999; Pearl, 1988).
The power of the hidden Markov model stems from the fact that
inference in the model is effective. Given a series of observations, o1, ..., on,
the probability distribution of the location at any given time can be
computed in order O(n) operations using the standard probabilistic
machinery developed for graphical models. Furthermore, maintaining the
distribution of the current location, as observations are made one by one, can
be done iteratively such that for each new observation, only constant, O(1),
time is needed. However, one should be cautious about the multiplicative
factors hidden in the O(n) and O(1) notation. We return to this issue shortly
below. Other possible inferences include tracking with a k step lag, i.e.,
maintaining the distribution of the location variable lt-k instead of the most
recent location, lt. This is called smoothing as the evolution of the location
variable lt-k as a function of time t is smoother than the evolution of the
current location lt. The Viterbi algorithm gives the most likely trajectory
given a sequence of observations, see (Rabiner, 1989).
In order to apply the hidden Markov model, one needs to specify two
kinds of probabilities. First, one needs to determine the conditional
11. Probabilistic Methods for Location Estimation in Wireless
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probability distribution of the observation variable given the state variable.
This is exactly the aim of calibration as discussed in Section 2. Second, the
conditional distribution of each state st given the previous state st-1, called the
transition probability, has to be determined. The form of these two kinds of
conditional probability distributions depends on whether the location and
observation variables are continuous or discrete. A continuous linearGaussian model for both transitions and observations yields the well-known
Kalman filter and smoothing equations (Kalman, 1960).
In the discrete case, the probability distributions are represented as
probability tables, which for transition probabilities constitute an N × N
matrix where N equals the number of possible locations. In the general case,
the multiplicative factor in the O(n) and O(1) notation above for the
computational complexity of inference is at least as large as N2. Methods to
reduce the computational complexity of tracking and smoothing when using
discrete-valued location include the aforementioned clustering approach that
reduces the number of locations N. In addition, a large proportion of state
transition probabilities are usually extremely small or zero. In such a case the
transition probability matrix is sparse which can be exploited to essentially
reduce computational complexity. One can also resort to approximative
inference using, for instance, particle filtering techniques that try to focus
computation on areas of the state space where most of the probability mass
lies (Fox et al., 1999).
7.
CONCLUSIONS
We showed how the probabilistic modelling approach can be used for
defining a unifying framework offering a theoretically solid solution to the
location estimation problem, and what is more, also to many related,
practically important problems involving issues like calibration, active
learning, error estimation and tracking with history. Nevertheless, having
said that, it must be acknowledged that problems in the real world are always
more complicated than the textbook examples, and developing these
theoretically elegant solutions to a robust, off-the-shelf software package
like for example the Ekahau Positioning Engine (see www.ekahau.com),
requires several minor but practically important technical tricks the details of
which are outside the scope of this paper. However, we strongly believe that
the best way to develop location-aware applications is to start with a
theoretically correct, “ideal” solution, and then approximate that solution as
accurately as possible given the pragmatic constraints defined by the realworld environment. Our experiences suggest that although it is not the only
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possible approach for this, the probabilistic modeling framework offers a
viable solution for developing practical applications in this domain.
ACKNOWLEDGEMENTS
This work was supported in part by the Academy of Finland under the
projects Cepler and Minos, and in part by the IST Programme of the
European Community, under the PASCAL Network of Excellence, IST2002-506778. This publication only reflects the authors' views.
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